Logarithmic Dynamics and Aggregation in Epidemics

Abstract

We consider a class of epidemiological models with an arbitrary number of infected compartments. We show that the logarithmic derivatives of the infected states converge to a consensus; this property rigorously explains the feature empirically observed in real epidemic data: the logarithms of the state variables associated with infected categories tend to behave as "parallel lines". We introduce and characterise the class of contagion functions, i.e., linear co-positive functions of the state variables that decrease (resp. increase) when the reproduction number is smaller (resp. larger) than 1. Finally, we analyse the generalised epidemiological model by considering the susceptible state variable along with a variable that aggregates all the infected compartments: this leads to an auxiliary planar system, governed by two differential inclusions, which has the same structure as the two-dimensional SI model and whose coefficients are functions of the original variables. We prove that well known properties of the classical SI model still hold in this generalised case.